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Quantum Mechanics: Time Independent Schrodinger Wave Equation

By Dragica Vasileska1, Gerhard Klimeck2

1. Arizona State University 2. Purdue University

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Abstract

In physics, especially quantum mechanics, the Schrödinger equation is an equation that describes how the quantum state of a physical system changes in time. It is as central to quantum mechanics as Newton's laws are to classical mechanics.

In the standard interpretation of quantum mechanics, the quantum state, also called a wavefunction or state vector, is the most complete description that can be given to a physical system. Solutions to Schrödinger's equation describe atomic and subatomic systems, electrons and atoms, but also macroscopic systems, possibly even the whole universe. The equation is named after Erwin Schrödinger who discovered it in 1926.

Schrödinger's equation can be mathematically transformed into the Heisenberg formalism, and into the Feynman path integral. The Schrödinger equation describes time in a way that is inconvenient for relativistic theories, a problem which is less severe in Heisenberg's formulation and completely absent in the path integral.

For stationary potential, the time-dependent Schrodinger equation reduces to the time-independent Schrodinger wave equation (TISWE). The TISWE can be solved for two types of problems: (1) open systems and (2) bound states. Reading material regarding the treatment of the open systems and the bound-state problem is provided in the link below. Also provided are links to the PCPBS Lab (piece-wise constant potential barrier system)and the BSP Lab (bound states problem). These two simulation labs, in addition to supplemental reading material, also contain homework exercises.

  • Reading Material: TISWE
  • Piece-Wise-Constant Potential Barrier Lab
  • Bound State Problem Lab
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    NSF

    Cite this work

    Researchers should cite this work as follows:

    • www.eas.asu.edu/~vasilesk
    • Dragica Vasileska; Gerhard Klimeck (2008), "Quantum Mechanics: Time Independent Schrodinger Wave Equation," https://nanohub.org/resources/4937.

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    In This Series

    1. Reading Material: Time Independent Schrodinger Wave Equation (TISWE)

      07 Jul 2008 | Teaching Materials | Contributor(s): Dragica Vasileska

      www.eas.asu.edu/~vasileskNSF

    2. Piece-Wise Constant Potential Barriers Tool

      30 Jun 2008 | Tools | Contributor(s): Xufeng Wang, Samarth Agarwal, Gerhard Klimeck, Dragica Vasileska, Mathieu Luisier, Jean Michel D Sellier

      Transmission and the reflection coefficient of a five, seven, nine, eleven and 2n-segment piece-wise constant potential energy profile

    3. Bound States Calculation Lab

      05 Jul 2008 | Tools | Contributor(s): Dragica Vasileska, Gerhard Klimeck, Xufeng Wang

      Calculates bound states for square, parabolic, triangular and V-shaped potential energy profile

    nanoHUB.org, a resource for nanoscience and nanotechnology, is supported by the National Science Foundation and other funding agencies. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.